(CNN) -- John Mark Karr, the man facing charges in the slaying of 6-year-old JonBenet Ramsey 10 years ago, told an officer in Thailand last week that he had sex with the girl before she died but added "there was no penetration," Thai Immigration Chief Gen. Suwat Tumrongsiskul told CNN on Wednesday.
"Karr then said, 'There are so many ways to have sex,' then he went quiet," Suwat said.
Karr's disclosure to the officer on duty was the only time the suspect mentioned sex involving the 6-year-old beauty pageant contestant during the four days he was held in Thai custody, Suwat said.
JonBenet's beaten and strangled body was found in the basement of the family home in Boulder, Colorado, the day after Christmas in 1996.
However, despite the incriminating statements Karr made to reporters and law enforcement officials in Bangkok last week, questions have surfaced as to whether the slight, soft-spoken schoolteacher could have been involved in the grisly killing of JonBenet.
Karr's immediate family strongly believes he had no involvement in the young girl's murder, according to family spokesman and attorney Gary Harris in Clayton, Georgia.
Harris, who is not representing John Karr, told CNN in a phone interview that family members do not remember Karr ever missing a Christmas gathering prior to his 2001 estrangement from them.
The family members Harris is representing said they would have remembered if Karr was not with his family because Karr was not financially well-off, Harris said. The family would have remembered if he had taken a plane trip somewhere.
Harris said later Tuesday he would turn over to Boulder Police a photograph taken at Christmas of 1996, showing Karr's three sons and another child. Karr is not seen in the photograph. The attorney said family members asserted that if Karr's three children were there, then Karr was as well.
Karr's second ex-wife, Lara Knutson, told a San Francisco television station last week that he was with her and the children during the entire 1996 Christmas season in Alabama.
However, according to Knutson's attorney, Michael Rains, "Lara has not located a photograph which places Mr. Karr either at their home in Hamilton, Alabama, or in any other location on or about Christmas Day 1996."
Knutson has turned over documents to Boulder County prosecutors, Rains said in a statement issued Tuesday, and continues to search for other materials that may be relevant. (Watch an examination of Karr's bizarre past -- 4:00)
When asked why Karr would make up a story about being involved in the Ramsey murder if he was actually in Alabama at the time, Harris said there may be a "problem with his emotional condition."
Still, a U.S. law enforcement source told CNN that Karr provided investigators with gruesome physical details about the condition of the girl's body -- details that had not been publicly disclosed.
While being transported to the Los Angeles County Jail, Karr made additional statements about JonBenet's death in the presence of officers, according to a source involved with Karr's transfer.
The source told CNN Karr began talking calmly to himself while looking out the window. He said, "Everybody says I couldn't know my way around the house, but I got in the house around 5 o'clock ... and I stayed there all night," according to the source. Karr also said that "they" -- meaning the Ramseys -- did not come back until about 10 p.m. He also repeated what he earlier told reporters -- that JonBenet's death was an accident, and that he was with her when she died.
Karr, 41, appeared Tuesday morning at a brief hearing in Los Angeles and waived his right to contest the extradition. He was represented by Haydeh Takasugi of the Los Angeles County Public Defender's Office. (Full story)
Karr's only words during the hearing were "Yes, your honor," when he was asked procedural questions by the judge. (Watch Karr in court -- 2:22)
The date of Karr's extradition to Colorado has not been released. The judge allowed 10 days for the transfer to take place.
Karr's being held on a warrant for suspicion of first-degree murder, kidnapping and child sexual assault, though he has not yet been directly charged with any crime.